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Engineering
81 Views   |

A triangle with vertices (4, 0), (–1, –1), (3, 5) is

• Option 1)

isosceles and right angled

• Option 2)

isosceles but not right angled

• Option 3)

right angled but not isosceles

• Option 4)

neither right angled nor isosceles

2
Engineering
129 Views   |

The equation of a circle with origin as a centre and passing through equilateral triangle whose median is of length $\dpi{100} 3a$ is

• Option 1)

$x^{2}+y^{2}=9a^{2}$

• Option 2)

$x^{2}+y^{2}=16a^{2}$

• Option 3)

$x^{2}+y^{2}=4a^{2}$

• Option 4)

$x^{2}+y^{2}=a^{2}$

As we learnt in Equation of a circle - - wherein Circle with centre and radius .  AD=3a AO =2a Centroid divides median in 2 : 1 Hence equation is    Option 1) This option is incorrect. Option 2) This option is incorrect. Option 3) This option is correct. Option 4) This option is incorrect.
Engineering
105 Views   |

The centre of the circle passing through  (0, 0) and (1, 0) and touching the circle $\dpi{100} x^{2}+y^{2}=9$   is

• Option 1)

$\left ( \frac{1}{2},\frac{1}{2} \right )\;$

• Option 2)

$\; \left ( \frac{1}{2},-\sqrt{2} \right )\;$

• Option 3)

$\; \left ( \frac{3}{2},\frac{1}{2} \right )\;$

• Option 4)

$\; \left ( \frac{1}{2},\frac{3}{2} \right )\;$

As we learnt in Common tangents of two circles - When two circles touch each other internally, there is only one common tangent. - wherein     Also  Point P is    Option 1) This option is incorrect. Option 2) This option is correct. Option 3) This option is incorrect. Option 4) This option is incorrect.
Engineering
99 Views   |

Locus of mid point of the portion between the axes of $\dpi{100} x\; \cos \alpha +y\; \sin \alpha =p\; where\; p$  is constant is

• Option 1)

$x^{2}+y^{2}=\frac{4}{p^{2}}\;$

• Option 2)

$\; x^{2}+y^{2}=4p^{2}\;$

• Option 3)

$\; \frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{2}{p^{2}}\;$

• Option 4)

$\; \; \frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{4}{p^{2}}$

As we learnt in  Normal form - - wherein p is the length of perpendicular segment from origin and  is the angle made by this perpendicular with +ve -axis.     Point A is:   Point B is:  We have, Option 1) This option is incorrect Option 2) This option is incorrect Option 3) This option is incorrect Option 4) This option is correct
Engineering
129 Views   |

The point of lines represented by  $\dpi{100} 3ax^{2}+5xy+(a^{2}-2)y^{2}=0$   and $\dpi{100} \perp$  to each other for

• Option 1)

two values of $a$

• Option 2)

$\forall \; \; a$

• Option 3)

for one value of $a$

• Option 4)

for no values of $a$

As we learnt in  General equation of a conic - - wherein   are constants     For perpendicular lines, sum of coefficients of x and y=0 So, there are two real values of a. Option 1) two values of     This option is correct Option 2) This option is incorrect Option 3) for one value of This option is incorrect Option 4) for no values of This option is incorrect
Engineering
126 Views   |

The centres of a set of circles, each of radius 3, lie on the circle $\dpi{100} x^{2}+y^{2}=25.$ The locus of any point in the set is

• Option 1)

$4\leq x^{2}+y^{2}\leq 64\;$

• Option 2)

$\; x^{2}+y^{2}\leq 25\;$

• Option 3)

$\; \; x^{2}+y^{2}\geq 25\;$

• Option 4)

$\; 3\leq x^{2}+y^{2}\leq 9$

As we learnt in  Common tangents of two circle - Where two circle neither intersect nor touch each other, there are 4 common tangents.Two are transverse and two are direct common tangents. - wherein     and     Common tangents of two circle - When they intersect, there are two common tangents, both of them being direct. - wherein     and     Common tangents of two circles - When two circles...
Engineering
152 Views   |

If the chord  $\dpi{100} y=mx+1$ of the circle $\dpi{100} x^{2}+y^{2}=1$   subtends an angle of measure 45° at the major segment of the circle then value of m is

• Option 1)

$2\pm \sqrt{2}$

• Option 2)

$-2\pm \sqrt{2}$

• Option 3)

$-1\pm \sqrt{2}$

• Option 4)

none of these

As we learnt in  Condition of tangency -   - wherein If    is a tangent to the circle     Equation of circle x2+y2=1 Now, y=mx+1 which represents pair of lines with On solving   Option 1) This option is incorrect Option 2) This option is incorrect Option 3) This option is correct Option 4) none of these This option is incorrect
Engineering
104 Views   |

Two common tangents to the circle $\dpi{100} x^{2}+y^{2}=2a^{2}$  and parabola $\dpi{100} y^{2}=8ax\; are$

• Option 1)

$x=\pm (y+2a)$

• Option 2)

$y=\pm (x+2a)$

• Option 3)

$x=\pm (y+a)$

• Option 4)

$y=\pm (x+a)$

As we learnt in  Equation of tangent - - wherein Tengent to is slope form.    and     Condition of tangency -   - wherein If    is a tangent to the circle     Tangent to circle x2+y2=2a2 and Tangent to parabola y2=8ax   Option 1) This option is incorrect Option 2) This option is correct Option 3) This option is incorrect Option 4) This option is incorrect
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