If the chord of contact of P(x_1, y_1) with r | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The equation of the chord of contact of $P(x_1, y_1)$ with respect to the circle $x^2+y^2=a^2$ is given by $xx_1 + yy_1 = a^2$. Let the circle be

Vedclass · Accepted Answer

$2a^2$

Vedclass · Answer

(B) The equation of the chord of contact of $P(x_1, y_1)$ with respect to the circle $x^2+y^2=a^2$ is given by $xx_1 + yy_1 = a^2$. Let the circle be $S: x^2+y^2-a^2=0$. The equation of the pair of tangents from $P$ to the circle is $SS_1 = T^2$,where $S_1 = x_1^2+y_1^2-a^2$ and $T = xx_1+yy_1-a^2$. This simplifies to $(x^2+y^2-a^2)(x_1^2+y_1^2-a^2) = (xx_1+yy_1-a^2)^2$. Since $\angle AOB = 90^{\circ}$,the lines $OA$ and $OB$ are perpendicular. For a circle $x^2+y^2+2gx+2fy+c=0$,the condition for perpendicular lines at the origin is $coeff(x^2) + coeff(y^2) = 0$. Expanding the equation: $(x^2+y^2-a^2)(x_1^2+y_1^2-a^2) = x^2x_1^2 + y^2y_1^2 + a^4 + 2xxy_1y - 2x^2x_1a^2 - 2yy_1a^2$. Sum of coefficients of $x^2$ and $y^2$ is $(x_1^2+y_1^2-a^2) - x_1^2 + (x_1^2+y_1^2-a^2) - y_1^2 = 0$. $x_1^2+y_1^2-a^2-x_1^2 + x_1^2+y_1^2-a^2-y_1^2 = 0$. $x_1^2+y_1^2-2a^2 = 0$,which gives $x_1^2+y_1^2 = 2a^2$.

If the chord of contact of $P(x_1, y_1)$ with respect to the circle $x^2+y^2=a^2$ meets the circle at $A$ and $B$; and if $\angle AOB=90^{\circ}$,then $x_1^2+y_1^2=$

Similar Questions

Let the tangent to the circle $C_{1}: x^{2}+y^{2}=2$ at the point $M(-1, 1)$ intersect the circle $C_{2}: (x-3)^{2}+(y-2)^{2}=5$ at two distinct points $A$ and $B$. If the tangents to $C_{2}$ at the points $A$ and $B$ intersect at $N$,then the area of the triangle $ANB$ is equal to

Given the circles $x^2 + y^2 - 4x - 5 = 0$ and $x^2 + y^2 + 6x - 2y + 6 = 0$. Let $P$ be a point $(\alpha, \beta)$ such that the lengths of the tangents from $P$ to both circles are equal. Then:

If the circle $x^2+y^2+2x+3y+1=0$ cuts another circle $x^2+y^2+4x+3y+2=0$ at points $A$ and $B$,then the equation of the circle with $AB$ as a diameter is

If $A$ and $B$ are the points of contact of the tangents drawn from the point $P(-3, 1)$ to the circle $x^2+y^2-4x+2y-4=0$,then the equation of the circumcircle of the triangle $PAB$ is

If tangents are drawn to the circle $x^2+y^2=12$ at the points of intersection with the circle $x^2+y^2-5x+3y-2=0$,then the ordinate of the point of intersection of these tangents is

Vedclass Test Series

Exam Paper Generator

Online Exam Module