The equation of the circle touching the circle $$ x^2 + y^2 - 6x + 6y + 17 = 0 $$ externally and to which the lines $$ x^2 - 3xy - 3x + 9y = 0 $$ are normal is:
Show Hint
- \( x^2 + y^2 - 3x + 2y - 2 = 0 \)
- \( x^2 + y^2 - 6x - 2y + 1 = 0 \)
- \( x^2 + y^2 - 6x - 2y - 1 = 0 \)
- \( x^2 + y^2 - 9x - 3y + 2 = 0 \)
The Correct Option is B
Solution and Explanation
Step 1: Finding the center and radius of the given circle Rewriting the given equation, \[ x^2 + y^2 - 6x + 6y + 17 = 0 \] Completing the square: \[ (x - 3)^2 - 9 + (y + 3)^2 - 9 + 17 = 0 \] \[ (x - 3)^2 + (y + 3)^2 = 1 \] So, the center is \( (3, -3) \) and radius \( R = 1 \).
Step 2: Finding the required circle The required circle is externally tangent, meaning its center lies along the normal lines. Using the given normal line condition, we solve for the appropriate equation: \[ x^2 + y^2 - 6x - 2y + 1 = 0. \]
Top Questions on circle
- Let circle $C$ be the image of
$$ x^2 + y^2 - 2x + 4y - 4 = 0 $$
in the line
$$ 2x - 3y + 5 = 0 $$
and $A$ be the point on $C$ such that $OA$ is parallel to the x-axis and $A$ lies on the right-hand side of the centre $O$ of $C$.
If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $\frac{1}{6}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to: - The maximum area of a rectangle inscribed in a circle of diameter \( R \) is:
- From a point A(0,3) on the circle \[ (x + 2)^2 + (y - 3)^2 = 4 \] a chord AB is drawn and extended to a point Q such that AQ = 2AB. Then the locus of Q is:
- If \( p \) and \( q \) be the longest and the shortest distance respectively of the point (-7,2) from any point (\(\alpha, \beta\)) on the curve whose equation is \[ x^2 + y^2 - 10x - 14y - 51 = 0 \] then the geometric mean (G.M.) of \( p \) is:
- The locus of the mid-point of a chord of the circle $x^2 + y^2 = 4$ which subtends a right angle at the origin is:
Questions Asked in AP EAMCET exam
Which of the following are ambident nucleophiles?
[A.] CN$^{\,-}$
[B.] CH$_{3}$COO$^{\,-}$
[C.] NO$_{2}^{\,-}$
[D.] CH$_{3}$O$^{\,-}$
[E.] NH$_{3}$- AP EAMCET - 2024
- Acids and Bases
- The correct sequence of enzymes involved in the commercial production of ethanol by fermentation from sugar is:
- AP EAMCET - 2024
- Acids and Bases
- At 298 K, the ionization constant of \( {CN}^- \) is \( 2.08 \times 10^{-6} \). What is the ionization constant of its conjugate acid? (Given \( K_w = 10^{-14} \))
- AP EAMCET - 2024
- Acids and Bases
Identify the anomers from the following.
- AP EAMCET - 2024
- Acids and Bases
The standard Gibbs free energy change \( \Delta G^\circ \) of a cell reaction is \(-301 { kJ/mol}\). What is \( E^\circ \) in volts?
(Given: \( F = 96500 { C/mol}\), \( n = 2 \))- AP EAMCET - 2024
- Acids and Bases